Pagina:Werkecarlf03gausrich.djvu/62

aequatio modo inventa doceat, functionis saltem terminum ultimum non evanescere. Supponemus, terminum altissimum functionis $f(w,L',L'',L'''{\text{ etc.}}),$ qui quidem coëfficientem non evanescentem habeat, esse $Nw^{\nu }$. Si igitur substituimus $w=x-a,$ patet, $f(x-a,L',L'',L'''{\text{ etc.}})$ esse functionem integram indeterminatarum $x,$ $a,$ sive quod idem est, functionem ipsius $x$ cum coëfficientibus ab indeterminata $a$ pendentibus, ita tamen ut terminus altissimus sit $Nx^{\nu }$, et proin coëfficientem determinatum ab $a$ non pendentem habeat, qui non sit $=0.$ Perinde $f(x-b,L',L'',L'''{\text{ etc.}}),$ $f(x-c,L',L'',L'''{\text{ etc.}})$ erunt functiones integrae indeterminatae $x,$ tales ut singularum terminus altissimus sit $Nx^{\nu }$, terminorum sequentium autem coëfficientes resp. $a,$ $b,$ $c$ etc. pendeant. Hinc productum ex $m$ factoribus

$${\begin{array}{c}f(x-a,L',L'',L'''{\text{ etc.}})\\f(x-b,L',L'',L'''{\text{ etc.}})\\f(x-c,L',L'',L'''{\text{ etc.}})\\{\text{etc.}}\end{array}}$$

erit functio integra ipsius $x,$ cuius terminus altissimus erit $N^{n}x^{m\nu },$ dum terminorum sequentium coëfficientes pendent ab indeterminatis $a,$ $b,$ $c$ etc.

Consideremus iam porro productum ex $m$ factoribus his

$${\begin{array}{c}f(x-a,l',l'',l'''{\text{ etc.}})\\f(x-b,l',l'',l'''{\text{ etc.}})\\f(x-c,l',l'',l'''{\text{ etc.}})\\{\text{etc.}}\end{array}}$$

quod quum sit functio indeterminatarum, $x,$ $a,$ $b,$ c etc. $l',$ $l'',$ $l'''$ etc., et quidem symmetrica respectu ipsarum $a,$ $b,$ $c$ etc., exhiberi poterit tamquam functio indeterminatarum $x,$ $\lambda ',$ $\lambda '',$ $\lambda '''$ etc. $l',$ $l'',$ $l'''$ etc. per

$${\varphi }(x,\lambda ',\lambda '',\lambda '''{\text{ etc., }}l',l'',l'''{\text{ etc.}})$$

denotanda. Erit itaque

$${\varphi }(x,\lambda ',\lambda '',\lambda '''{\text{ etc., }}\lambda ',\lambda '',\lambda '''{\text{ etc.}})$$

productum ex factoribus

$${\begin{array}{c}f(x-a,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\f(x-b,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\f(x-c,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\{\text{etc.}}\end{array}}$$